Abstract

New results concerning flow velocity and solute spreading in an unbounded three-dimensional partially saturated heterogeneous porous formation are derived. Assuming that the effective water content is a uniformly distributed constant, and dealing with the recent results of Severino and Santini (Advances in Water Resources 2005;28:964–974) on mean vertical steady flows, first-order approximation of the velocity covariance , and concurrently of the resultant macrodispersion coefficients are calculated. Generally, the velocity covariance is expressed via two quadratures. These quadratures are further reduced after adopting specific (i.e., exponential) shape for the required (cross)correlation functions. Two particular formation structures that are relevant for the applications and lead to significant simplifications of the computational aspect are also considered.

It is shown that the rate at which the Fickian regime is approached is an intrinsic medium property, whereas the value of the macrodispersion coefficients is also influenced by the mean flow conditions as well as the (cross)variances σ2γ of the input parameters. For a medium of given anisotropy structure, the velocity variances reduce as the medium becomes drier (in mean), and it increases with σ2γ. In order to emphasize the intrinsic nature of the velocity autocorrelation, good agreement is shown between our analytical results and the velocity autocorrelation as determined by Russo (Water Resources Research 1995;31:129–137) when accounting for groundwater flow normal to the formation bedding. In a similar manner, the intrinsic character of attainment of the Fickian regime is demonstrated by comparing the scaled longitudinal macrodispersion coefficients \(\frac{D_{11} (t)}{{D_{11}}^{(\infty)}}\) as well as the lateral displacement variance \(\frac{X_{22}(t)}{{X_{22}}^{(\infty)}} = \frac{X_{33} (t)}{{X_{33}}^{(\infty)}}\) with the same quantities derived by Russo (Water Resources Research 1995;31:129–137) in the case of groundwater flow normal to the formation bedding.

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